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In the present paper, we first establish and verify a new sharp hyperbolic version of the Michael-Simon inequality for mean curvatures in hyperbolic space H n + 1 based on the locally constrained inverse curvature flow introduced by Brendle, Guan and Li ("An inverse curvature type hypersurface flow in H n + 1 ," (Preprint)) as follows (0.1) ∫ M λ ′ f 2 E 1 2 + | ∇ M f | 2 − ∫ M ∇ ̄ f λ ′ , ν + ∫ ∂ M f ≥ ω n 1 n ∫ M f n n − 1 n − 1 n provided that M is h-convex and f is a positive smooth function, where λ′(r) = coshr. In particular, when f is of constant, (0.1) coincides with the Minkowski type inequality stated by Brendle, Hung, and Wang in ("A Minkowski inequality for hypersurfaces in the anti-de Sitter-Schwarzschild manifold," Commun. Pure Appl. Math., vol. 69, no. 1, pp. 124–144, 2016). Further, we also establish and confirm a new sharp Michael-Simon inequality for the kth mean curvatures in H n + 1 by virtue of the Brendle-Guan-Li's flow ("An inverse curvature type hypersurface flow in H n + 1 ," (Preprint)) as below (0.2) ∫ M λ ′ f 2 E k 2 + | ∇ M f | 2 E k − 1 2 − ∫ M ∇ ̄ f λ ′ , ν ⋅ E k − 1 + ∫ ∂ M f ⋅ E k − 1 ≥ p k ◦ q 1 − 1 ( W 1 (Ω)) 1 n − k + 1 ∫ M f n − k + 1 n − k ⋅ E k − 1 n − k n − k + 1 provided that M is h-convex and Ω is the domain enclosed by M, pk(r) = ωn(λ′)k−1, W 1 (Ω) = 1 n | M | , λ′(r) = coshr, q 1 (r) = W 1 S r n + 1 , the area for a geodesic sphere of radius r, and q 1 − 1 is the inverse function of q1. In particular, when f is of constant and k is odd, (0.2) is exactly the weighted Alexandrov–Fenchel inequalities proven by Hu, Li, and Wei in ("Locally constrained curvature flows and geometric inequalities in hyperbolic space," Math. Ann., vol. 382, nos. 3–4, pp. 1425–1474, 2022). [ABSTRACT FROM AUTHOR]

In the present paper, we first establish and verify a new sharp hyperbolic version of the Michael-Simon inequality for mean curvatures in hyperbolic space Hn+1 ${\mathbb{H}}^{n+1}$ based on the locally constrained inverse curvature flow introduced by Brendle, Guan and Li (“An inverse curvature type hypersurface flow in Hn+1 ${\mathbb{H}}^{n+1}$ ,” (Preprint)) as follows(0.1)∫Mλ′f2E12+|∇Mf|2−∫M∇̄fλ′,ν+∫∂Mf≥ωn1n∫Mfnn−1n−1n $$\underset{M}{\int }{\lambda }^{\prime }\sqrt{{f}^{2}{E}_{1}^{2}+\vert {\nabla }^{M}f{\vert }^{2}}-\underset{M}{\int }\langle \bar{\nabla }\left(f{\lambda }^{\prime }\right),\nu \rangle +\underset{\partial M}{\int }f\ge {\omega }_{n}^{\frac{1}{n}}{\left(\underset{M}{\int }{f}^{\frac{n}{n-1}}\right)}^{\frac{n-1}{n}}$$ provided that M is h-convex and f is a positive smooth function, where λ′(r) = coshr. In particular, when f is of constant, (0.1) coincides with the Minkowski type inequality stated by Brendle, Hung, and Wang in (“A Minkowski inequality for hypersurfaces in the anti-de Sitter-Schwarzschild manifold,” Commun. Pure Appl. Math., vol. 69, no. 1, pp. 124–144, 2016). Further, we also establish and confirm a new sharp Michael-Simon inequality for the kth mean curvatures in Hn+1 ${\mathbb{H}}^{n+1}$ by virtue of the Brendle-Guan-Li’s flow (“An inverse curvature type hypersurface flow in Hn+1 ${\mathbb{H}}^{n+1}$ ,” (Preprint)) as below(0.2)∫Mλ′f2Ek2+|∇Mf|2Ek−12−∫M∇̄fλ′,ν⋅Ek−1+∫∂Mf⋅Ek−1≥pk◦q1−1(W1(Ω))1n−k+1∫Mfn−k+1n−k⋅Ek−1n−kn−k+1 \begin{align}\hfill & \underset{M}{\int }{\lambda }^{\prime }\sqrt{{f}^{2}{E}_{k}^{2}+\vert {\nabla }^{M}f{\vert }^{2}{E}_{k-1}^{2}}-\underset{M}{\int }\langle \bar{\nabla }\left(f{\lambda }^{\prime }\right),\nu \rangle \cdot {E}_{k-1}+\underset{\partial M}{\int }f\cdot {E}_{k-1}\hfill \\ \hfill & \quad \ge {\left({p}_{k}{\circ}{q}_{1}^{-1}\left({W}_{1}\left({\Omega}\right)\right)\right)}^{\frac{1}{n-k+1}}{\left(\underset{M}{\int }{f}^{\frac{n-k+1}{n-k}}\cdot {E}_{k-1}\right)}^{\frac{n-k}{n-k+1}}\hfill \end{align} provided that M is h-convex and Ω is the domain enclosed by M, p k(r) = ω n(λ′)k−1, W1(Ω)=1n|M| ${W}_{1}\left({\Omega}\right)=\frac{1}{n}\vert M\vert $ , λ′(r) = coshr, q1(r)=W1Srn+1 ${q}_{1}\left(r\right)={W}_{1}\left({S}_{r}^{n+1}\right)$ , the area for a geodesic sphere of radius r, and q1−1 ${q}_{1}^{-1}$ is the inverse function of q 1. In particular, when f is of constant and k is odd, (0.2) is exactly the weighted Alexandrov–Fenchel inequalities proven by Hu, Li, and Wei in (“Locally constrained curvature flows and geometric inequalities in hyperbolic space,” Math. Ann., vol. 382, nos. 3–4, pp. 1425–1474, 2022).

Assume that k ≤ d is a positive integer and 퓒 is a finite collection of convex bodies in ℝd. We prove a Helly-type theorem: If for every subfamily 퓒* ⊂ 퓒 of size at most max{d + 1, 2(d – k + 1)} the set ⋂ 퓒* contains a k-dimensional cone, then so does ⋂ 퓒. One ingredient in the proof is another Helly-type theorem about the dimension of lineality spaces of convex cones. [ABSTRACT FROM AUTHOR]

Given a strictly increasing continuous function ϕ : R ≥ 0 ⟶ R ∪ { - ∞ } with lim t → ∞ ϕ (t) = ∞ , a function f : [ a , b ] ⟶ R ≥ 0 is said to be ϕ -concave if ϕ ∘ f is concave. When ϕ (t) = t p , p > 0 , this notion is that of p-concavity whereas for ϕ (t) = log (t) it leads to the so-called log-concavity. In this work, we show that if the cross-sections volume function of a compact set K ⊂ R n (of positive volume) w.r.t. some hyperplane H passing through its centroid is ϕ -concave, then one can find a sharp lower bound for the ratio vol (K -) / vol (K) , where K - denotes the intersection of K with a halfspace bounded by H. When K is convex, this inequality recovers a classical result by Grünbaum. Moreover, other related results for ϕ -concave functions (and involving the centroid) are shown. [ABSTRACT FROM AUTHOR]

The Blaschke–Leichtweiss theorem (Leichtweiss in Abh Math Semin Univ Hambg 75:257–284, 2005) states that the smallest area convex domain of constant width w in the 2-dimensional spherical space S 2 is the spherical Reuleaux triangle for all 0 < w ≤ π 2 . In this paper we extend this result to the family of wide r-disk domains of S 2 , where 0 < r ≤ π 2 . Here a wide r-disk domain is an intersection of spherical disks of radius r with centers contained in their intersection. This gives a new and elementary proof of the Blaschke–Leichtweiss theorem. Furthermore, we investigate the higher dimensional analogue of wide r-disk domains called wide r-ball bodies. In particular, we determine their minimum spherical width (resp., inradius) in the spherical d-space S d for all d ≥ 2 . Also, it is shown that any minimum volume wide r-ball body is of constant width r in S d , d ≥ 2 . [ABSTRACT FROM AUTHOR]

The Brouwer fixed point theorem is a key ingredient in the proof that a periodic differential equation has a periodic solution in a set that satisfies a suitable tangency condition on its boundary. The main goal of this note is to show that both results are in fact equivalent. [ABSTRACT FROM AUTHOR]

In this paper we study the L p Gauss image problem, which is a generalization of the L p Aleksandrov problem and the Gauss image problem in convex geometry. We obtain the existence result for the L p Gauss image problem in two cases (i) p > 0 or (ii) p < 0 with the given even measures. [ABSTRACT FROM AUTHOR]

The isoperimetric quotient of the whole family of inner and outer parallel bodies of a convex body is shown to be decreasing in the parameter of definition of parallel bodies, along with a characterization of those convex bodies for which that quotient happens to be constant on some interval within its domain. This is obtained relative to arbitrary gauge bodies, having the classical Euclidean setting as a particular case. Similar results are established for different families of Wulff shapes that are closely related to parallel bodies. These give rise to solutions of isoperimetric-type problems. Furthermore, new results on the monotonicity of quotients of other quermassintegrals different from surface area and volume, for the family of parallel bodies, are obtained. [ABSTRACT FROM AUTHOR]

Given a compact set K ⊂ R n of positive volume, and fixing a hyperplane H passing through its centroid, we find a sharp lower bound for the ratio vol (K -) / vol (K) , depending on the concavity nature of the function that gives the volumes of cross-sections (parallel to H) of K, where K - denotes the intersection of K with a halfspace bounded by H. When K is convex, this inequality recovers a classical result by Grünbaum. To this respect, we also show that the log-concave case is the limit concavity assumption for such a generalization of Grünbaum's inequality. [ABSTRACT FROM AUTHOR]

A classification of $${\text {SL}}(n)$$ invariant valuations on the space of convex polytopes in $$\mathbb {R}^n$$ without any continuity assumptions is established. A corresponding result is obtained on the space of convex polytopes in $$\mathbb {R}^n$$ that contain the origin. [ABSTRACT FROM AUTHOR]

An r-hyperconvex body is a set in the d-dimensional Euclidean space 피 d that is the intersection of a family of closed balls of radius r. We prove the analogue of the classical Blaschke-Santaló inequality for r-hyperconvex bodies, and we also establish a stability version of it. The other main result of the paper is an r-hyperconvex version of the reverse isoperimetric inequality in the plane. [ABSTRACT FROM AUTHOR]

We validate the conjectured logarithmic Minkowski inequality, and thus the equivalent logarithmic Brunn–Minkowski inequality, in some particular cases and we prove some variants of the logarithmic Minkowski inequality for general convex bodies without the symmetry assumption. An application of one of these variants is shown. [ABSTRACT FROM AUTHOR]

Let $$P$$ be a polytope and $$P_{\lambda }, \lambda \le 0$$ an inner parallel body of $$P$$ , i.e., the polytope constructed by moving the facets of $$P$$ inwards at distance $$\left| \lambda \right| $$ . We study when $$P_{\lambda }$$ is a summand of $$P$$ . We characterize the polytopes $$P$$ satisfying that $$P_{\lambda }$$ is a summand of $$P_{\mu }$$ with $$\lambda <\mu \le 0$$ . Besides, we provide an explicit decomposition of such polytopes using the so called form bodies of their inner parallel bodies. [ABSTRACT FROM AUTHOR]

The intrinsic volumes of a convex cone are geometric functionals that return basic structural information about the cone. Recent research has demonstrated that conic intrinsic volumes are valuable for understanding the behavior of random convex optimization problems. This paper develops a systematic technique for studying conic intrinsic volumes using methods from probability. At the heart of this approach is a general Steiner formula for cones. This result converts questions about the intrinsic volumes into questions about the projection of a Gaussian random vector onto the cone, which can then be resolved using tools from Gaussian analysis. The approach leads to new identities and bounds for the intrinsic volumes of a cone, including a near-optimal concentration inequality. [ABSTRACT FROM AUTHOR]

In this paper, we first investigate several rigidity problems for hypersurfaces in the warped product manifolds with constant linear combinations of higher order mean curvatures as well as 'weighted' mean curvatures, which extend the work (Brendle in Publ Math Inst Hautes Études Sci 117:247-269, ; Brendle and Eichmair in J Differ Geom 94(94):387-407, ; Montiel in Indiana Univ Math J 48:711-748, ) considering constant mean curvature functions. Secondly, we obtain the rigidity results for hypersurfaces in the space forms with constant linear combinations of intrinsic Gauss-Bonnet curvatures $$L_k$$ . To achieve this, we develop some new kind of Newton-Maclaurin type inequalities on $$L_k$$ which may have independent interest. [ABSTRACT FROM AUTHOR]

polynomial representation of a convex d-polytope P is a finite set { p( x), . . . , p( x)} of polynomials over $${\mathbb {R}^d}$$ such that $${P=\{x \in \mathbb {R}^d : p_i(x) \ge 0 \mbox{ for every }1 \le i \le n\}}$$. Let s( d, P) be the least possible n as above. It is conjectured that s( d, P) = d for all convex d-polytopes P. We confirm this conjecture for simple d-polytopes by providing an explicit construction of d polynomials that represent a given simple d-polytope P. [ABSTRACT FROM AUTHOR]

Suppose that a finite-dimensional cube is orthogonally projected onto a central section of itself by a subspace of one dimension less. Up to dimension 9, at least one vertex is projected onto the section, but for dimension 10 or larger, there are orthogonal projections for which all the vertices are projected outside the section. In fact, this is the case for "most" orthogonal projections, as the dimension tends to infinity. [ABSTRACT FROM AUTHOR]

For matrices with all nonnegative entries, the Perron–Frobenius theorem guarantees the existence of an eigenvector with all nonnegative components. We show that the existence of such an eigenvector is also guaranteed for a very different class of matrices, namely real symmetric matrices with exactly two eigenvalues. We also prove a partial converse, that among real symmetric matrices with any more than two eigenvalues there exist some having no nonnegative eigenvector. [ABSTRACT FROM AUTHOR]